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We study the semigroup$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ of monotoneinjective partial selfmaps of the set of$L_n\times_{\operatorname{lex}}\mathbb{Z}$ having co-finite domain and image,where $L_n\times_{\operatorname{lex}}\mathbb{Z}$ is the lexicographic productof $n$-elements chain and the set of integers with the usual order. We showthat $\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is bisimpleand establish its projective congruences. We prove that$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is finitelygenerated, and for $n=1$ every automorphism of$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ is inner andshow that in the case $n\geqslant 2$ the semigroup$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ has non-innerautomorphisms. Also we show that every Baire topology $\tau$ on$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ such that$(\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}}),\tau)$ is aHausdorff semitopological semigroup is discrete, construct a non-discreteHausdorff semigroup inverse topology on$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$, and prove thatthe discrete semigroup$\mathscr{I\!O}\!_{\infty}(\mathbb{Z}^n_{\operatorname{lex}})$ cannot beembedded into some classes of compact-like topological semigroups and that itsremainder under the closure in a topological semigroup $S$ is an ideal in $S$.

On monoids of monotone injective partial selfmaps of $L_n\times_{\operatorname{lex}}\mathbb{Z}$ with co-finite domains and images